Here we shall consider vector multiplets only and focus on an abelian gauging, following the conventions
of [1]}In order to be consistent with our conventions it is necessary to switch
the sign of \(C_{\alpha \beta \gamma }\)  and \(h^\alpha \)  and to take \(\varepsilon _{(5)}^{\mu \nu \rho \sigma z}=\varepsilon _{(4)}^{\mu \nu \rho \sigma }\) .. The dimensional reduction is performed along an \(S^1\)  with
compact coordinate \(z\)  by means of the Kaluza-Klein ansatz
\(\begin{gathered}ds_{(5)}^2 = e^\frac{\phi }{\sqrt{3}} ds_{(4)}^2 + e^{-\frac{2\phi }{\sqrt{3}}} (dz + K_\mu dx^\mu )^2\,,\qquad A^\alpha = B^\alpha (dz + K_\mu dx^\mu ) + C^\alpha _\mu dx^\mu \,, \\z^\alpha = e^{-\frac{\phi }{\sqrt{3}}} h^\alpha + i B^\alpha \,, \qquad e^\mathcal {K} = \frac{1}{8}e^{\sqrt{3}\phi }\,,\qquad F^I_{\mu \nu } = \frac{1}{\sqrt{2}} (dK_{\mu \nu }, dC^\alpha _{\mu \nu })\,,\end{gathered}\) 
